BCS |
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| The first widely-accepted theory to explain superconductivity put forth in 1957 by John Bardeen, Leon Cooper, and John Schreiffer. The theory asserts that, as electrons pass through a crystal lattice, the lattice deforms inward towards the electrons generating sound packets known as "phonons". These phonons produce a trough of positive charge in the area of deformation that assists subsequent electrons in passing through the same region in a process known as phonon-mediated coupling. This is analogous to rolling a bowling ball up the middle of a bed. 2 people, one lying on each side of the bed, will tend to roll toward the center of the bed, once the ball has created a depression in the mattress. And, a 2nd bowling ball, placed at the foot of the bed, will now, quite easily, roll toward the middle. For a more technical explanation click here. | |
| The properties of Type I
superconductors were modeled successfully by the efforts
of John Bardeen, Leon Cooper, and Robert Schrieffer in
what is commonly called the BCS theory. A key conceptual
element in this theory is the pairing of electrons close
to the Fermi level into Cooper pairs through interaction
with the crystal lattice. This pairing results from a
slight attraction between the electrons related to
lattice vibrations; the coupling to the lattice is called
a phonon interaction. Pairs of electrons can behave very differently from single electrons which are fermions and must obey the Pauli exclusion principle. The pairs of electrons act more like bosons which can condense into the same energy level. The electron pairs have a slightly lower energy and leave an energy gap above them on the order of .001 eV which inhibits the kind of collision interactions which lead to ordinary resistivity. For temperatures such that the thermal energy is less than the band gap, the material exhibits zero resistivity. Bardeen, Cooper, and Schrieffer received the Nobel Prize in 1972 for the development of the theory of superconductivity. |
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| The BCS theory of superconductivity
built on earlier work by Frolich and Cooper. Frolich's
work showed that given the right conditions, electrons
could experience an attractive interaction mediated by
phonons. Cooper then showed that given this attractive
interaction occurred, then if two electrons were to be
scattered from near the fermi surface to just above the
fermi surface, it would be energetically preferable for
the two electrons to form a pair with both electrons
above their former energy, than for the electrons to
return to their former state. Such a pair of electrons is
known as a Cooper pair. The total momentum of a Cooper
pair is constant, and the spins of the two electrons
forming the Cooper pair are opposite to each other. The BCS theory showed that it was possible for a number of Cooper pairs to form above the same fermi surface. These Cooper pairs form a homogenous condensate, and for a pair to leave the condensate, it must split apart into to separate electrons, which requires a certain amount of energy. The condensate has long range coherence, and it is this coherence that leads to superconductivity. If an electric field is applied across a superconductor, it is not possible for individual cooper pairs to move in response to the field, but the condensate must move as a whole. Provided that the electric field does not provide enough energy to split the Cooper pairs, the condensate will move throughout the medium without resistance. This also provides an understanding of why some of the best conductors do not superconduct. Conductors such as gold and platinum conduct well because the electron-phonon interaction is very weak. However because the electron-phonon interaction is very weak, the electron-electron interaction is never attractive, and so Cooper pairs cannot form, and thus superconductivity does not occur. In materials such as lead, which don't conduct as well as gold or platinum, but do superconduct, the electron-phonon interaction is much stronger, allowing the Cooper pairs to form. |
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| Ideas Leading to the BCS Theory | |
| The BCS theory of superconductivity
has successfully described the measured properties of
Type I superconductors. It envisions resistance-free
conduction of coupled pairs of electrons called Cooper
pairs. This theory is remarkable enough that it is
interesting to look at the chain of ideas which led to it.
One of the first steps toward a theory of superconductivity was the realization that there must be a band gap separating the charge carriers from the state of normal conduction. A band gap was implied by the very fact that the resistance is precisely zero. If charge carriers can move through a crystal lattice without interacting at all, it must be because their energies are quantized such that they do not have any available energy levels within reach of the energies of interaction with the lattice. A band gap is suggested by specific heats of materials like vanadium. The fact that there is an exponentially increasing specific heat as the temperature approaches the critical temperature from below implies that thermal energy is being used to bridge some kind of gap in energy. As the temperature increases, there is an exponential increase in the number of particles which would have enough energy to cross the gap. The critical temperature for superconductivity must be a measure of the band gap, since the material could lose superconductivity if thermal energy could get charge carriers across the gap. The critical temperature was found to depend upon isotopic mass. It certainly would not if the conduction was by free electrons alone. This made it evident that the superconducting transition involved some kind of interaction with the crystal lattice. Single electrons could be eliminated as the charge carriers in superconductivity since with a system of fermions you don't get energy gaps. All available levels up to the Fermi energy fill up. The decrease of the band gap with increase in temperature as you approach the critical temperature suggests a charge carrier with some collective properties. That is, it acts like something tied together with a bond which is weakened by the thermal interaction and destroyed at the critical temperature. The needed boson behavior was consistent with having coupled pairs of electrons with opposite spins. The isotope effect described above suggested that the coupling mechanism involved the crystal lattice, so this gave rise to the phonon model of coupling. |
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| Measured Superconductor Bandgap : | |
| The measured bandgap in Type I superconductors is one of the pieces of experimental evidence which supports the BCS theory. The BCS theory predicts a bandgap of | |
E = (7/2) * k * Tc |
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| The energy gap is related to the coherence length for the superconductor, one of the two characteristic lengths associated with superconductivity. | |
| The effective energy gap in
superconductors can be measured in microwave absorption
experiments. The data at left offer general confirmation
of the BCS theory of superconductivity. The data is
attributed to Townsend and Sutton. The reduction of the energy gap as you approach the critical temperature can be taken as an indication that the charge carriers have some sort of collective nature. That is, the charge carriers must consist of at least two things which are bound together, and the binding energy is weakening as you approach the critical temperature. Above the critical temperature, such collections do not exist, and normal resistivity prevails. This kind of evidence, along with the isotope effect which showed that the crystal lattice was involved, helped to suggest the picture of paired electrons bound together by phonon interactions with the lattice. |
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| Experimental Support: BCS Theory | |
| Electrons acting as pairs via
lattice interaction? How did they come up with that idea
for the BCS theory of superconductivity? The evidence for
a small band gap at the Fermi level was a key piece in
the puzzle. That evidence comes from the existence of a
critical temperature, the existence of a critical
magnetic field, and the exponential nature of the heat
capacity variation in the Type I superconductors. The evidence for interaction with the crystal lattice came first from the isotope effect on the critical temperature. The band gap suggested a phase transition in which there was a kind of condensation, like a Bose-Einstein condensation, but electrons alone cannot condense into the same energy level (Pauli exclusion principle). Yet a drastic change in conductivity demanded a drastic change in electron behavior. Perhaps coupled pairs of electrons with antiparallel spins could act like bosons? |
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